Examples of Stability of Tensor Products in Positive Characteristic

Abstract

Let X be projective smooth variety over an algebraically closed field k and let ℰ, ℱ be μ-semistable locally free sheaves on X. When the base field is ℂ, using transcendental methods, one can prove that the tensor product ℰ⊗ℱ is always a μ-semistable sheaf. However, this theorem is no longer true over positive characteristic; for an analogous theorem one needs the hypothesis of strong μ-semistability; nevertheless, this hypothesis is not a necessary condition. The objective of this paper is to construct, without the strongly μ-semistability hypothesis, a family of locally free sheaves with μ-stable tensor product.

1. Introduction

When the base field is ℂ, the Kobayashi-Hitchin correspondence ensures that a vector bundle on a complex projective variety is polystable if and only if it admits a Hermitian-Einstein metric with respect to the Kähler metric induced by 𝒪_𝑋(1) (see [1] for curves and [2] for complex compact varieties). In this way, one can prove that the tensor product of Hermitian-Einstein bundles is again Hermitian-Einstein, therefore polystable, and the same is true for symmetric and exterior products. However, in positive characteristic, this is false; in [3], Gieseker proved the existence of stable vector bundles on curves with nonsemistable symmetric products. When these bundles are of degree zero, the nonsemistability of the symmetric product imply the nonsemistability of the tensor product. One way to solve this problem is to introduce the concept of strong 𝜇-semistability. Let 𝐹∶𝑋→𝑋 be a total Frobenius morphism of 𝑋; we say that ℰ is strongly 𝜇-semistable if for all 𝑚≥0 the pullback (𝐹^𝑚)^∗ℰ is 𝜇-semistable with respect to the induced polarization (𝐹^𝑚)^∗𝒪_𝑋(1). Under these assumptions, the tensor product of strongly 𝜇-semistable bundles is again strongly 𝜇-semistable (see [4, Section 7] for curves and [5] for general case). However, in general, there are no conditions to ensure the 𝜇-semistability of a tensor product of 𝜇-semistable bundles (at least not known to the author).

The aim of this work is the construction of examples, in any characteristic, of families of 𝜇-stable bundles with 𝜇-stable tensor products, this without the assumption of strong 𝜇-stability. The key result is Proposition 3.1, which shows that, if 𝜋∶𝑌→𝑋 is étale and Galois with g.c.d.(|𝐺|,char(𝑘))=1, then ℱ is 𝜇-semistable (𝜇-polystable) if and only if 𝜋^∗ℱ is 𝜇-semistable (𝜇-polystable, resp.). We remark that, in positive characteristic, this result is false for an arbitrary finite morphism. Under these hypotheses, if ℒ is a line bundle on 𝑌, in Corollary 3.2 we show that ℱ⊗𝜋_∗ℒ is 𝜇-(semi)polystable if ℱ is 𝜇-(semi)stable. If in addition g.c.d.(degℱ,𝑟𝑘(ℱ))=1=(|𝐺|,𝑟𝑘(ℱ)) and 𝑔^∗ℒ≠ℒ for all 𝑔∈𝐺, then in Corollary 3.5 it is proved that ℱ⊗𝜋_∗ℒ is 𝜇-stable. The other examples that are constructed come from the 𝐺-invariant decomposition of 𝜋_∗𝒪_𝑌. Now, the general outline of this construction is as follows. In Section 1, we observe that 𝜋_∗𝒪_𝑌 has a natural 𝐺-invariant decomposition: 𝜋_∗𝒪_𝑌=_𝑉∈ℐ𝑉⊗_𝑘ℰ_𝑉,(1.1) where ℐ is the set of irreducible representations of 𝐺 over 𝑘 and each ℰ_𝑉 is a locally free sheaf of rank dim𝑉. Also, we have the relations ℰ_𝑉1⊗ℰ_𝑉2≃_𝑇∈ℐℰ^⊕𝑛𝑇_𝑇,∧^𝑠ℰ_𝑉≃_𝑊∈ℐℰ^⊕𝑛𝑊_𝑊,(1.2) where 𝑉⊗𝑊≃⨁_𝑇∈ℐ𝑇^⊕𝑛𝑇 and ∧^𝑠𝑉=⨁_𝑊∈ℐ𝑊^⊕𝑛𝑊. In Section 2, we show that ℰ_𝑉 is 𝜇-𝐻-stable for any polarization on 𝑋 and also that ℱ⊗ℰ_𝑉 is 𝜇-𝐻-polystable (𝜇-𝐻-semistable) whenever ℱ is 𝜇-𝐻-stable (𝜇-𝐻-semistable, resp.). Furthermore, let ℳ_𝑋(𝑟,𝑑) be the moduli space parametrizing 𝜇-𝐻-stable sheaves of rank 𝑟 and degree 𝑑, if g.c.d.(𝑑,𝑟)=1=g.c.d.(|𝐺|,𝑟), we proved in Theorem 3.12 that, for any irreducible representation 𝑉 of 𝐺 and ℱ∈ℳ_𝑋(𝑟,𝑑), ℱ⊗ℰ_𝑉 is 𝜇-𝐻-stable and the natural morphism ℳ_𝑋(𝑟,𝑑)→ℳ_𝑋(𝑟⋅dim𝑉,𝑑⋅dim𝑉) given by ℱ↦ℱ⊗ℰ_𝑉 is injective.

In Section 3, we proved that, if g.c.d.(𝑑,|𝐺|)=1, then, for all 𝑟>0, 𝐺 acts without fixed points on ℳ_𝑌(𝑟,𝑑); in particular the morphisms ℳ_𝑌(𝑟,𝑑)→ℳ_𝑌(𝑟,𝑑)/𝐺 are étale Galois covers, thus we have nontrivial examples for Section 2. Finally, when 𝑋 is an abelian variety and 𝜋∶𝑌→𝑋 is an isogeny of degree 𝑟, Proposition 4.5 shows that, if g.c.d(𝑟,𝑑)=1, then each irreducible component of Pic^𝑑(𝑌)/𝐺 is an irreducible component of ℳ_𝑋(𝑟,𝑑), where 𝐺=Ker(𝜋). Section 5 is devoted to prove Proposition 3.1.

1.1. Notations and Conventions

Throughout the paper, 𝑘 denotes an algebraically closed field and 𝐺 a finite group satisfying that g.c.d.(|𝐺|,char(𝑘))=1. Also, 𝑋 denotes a smooth projective variety over 𝑘 with a fixed polarization, that is, with a fixed ample line bundle 𝒪_𝑋(1), and we denote by 𝐻 any divisor in the linear system |𝒪_𝑋(1)|.

We denote by 𝑘[𝐺] the group algebra of 𝐺 with coefficients in 𝑘. Also, if 𝑇₁ and 𝑇₂ are representations of 𝐺 over 𝑘 then we denote by ⟨𝑇₁,𝑇₂⟩ the dimHom_𝑘(𝑇₁,𝑇₂)^𝐺=dimHom_𝑘[𝐺](𝑇₁,𝑇₂). We will identify vector bundles on 𝑋 with locally free 𝒪_𝑋-modules.

2. Étale Covers

Let 𝜋∶𝑌→𝑋 be an étale Galois cover. Now, we recall that a locally free sheaf ℱ on 𝑌 is a 𝐺-sheaf (see [6, page 69]) if 𝐺 acts on ℱ in a way compatible with the action on 𝑌. Since 𝜋 is étale, it is flat; hence, we have that 𝜋_∗ℱ is a locally free 𝒪_𝑋-module with an action of 𝐺. From this, we have defined a natural morphism of 𝑘-algebras 𝑘[𝐺]→End_𝒪𝑋(𝜋_∗ℱ). Now, as we suppose by hypothesis that g.c.d.(char(𝑘),|𝐺|)=1, Maschke's Theorem guarantees that 𝑘[𝐺] is a semisimple 𝑘-algebra of finite dimension |𝐺| over 𝑘. We denote by ℐ={𝑉₀,…,𝑉_𝑟} the set of irreducible representations of 𝐺 over 𝑘, that is, the set of irreducible 𝑘[𝐺]-modules, and by {𝑒_𝑉}_𝑉∈ℐ the set of corresponding idempotents. Thus, we have that 𝜋_∗ℱ=_𝑉∈ℐ𝑒_𝑉𝜋_∗ℱ.(2.1)

On the other hand, by Theorem 1(B) in [6, page 111], there is a locally free sheaf ℋ on 𝑋 such that ℱ≃𝜋^∗ℋ, so we have natural 𝐺-invariant isomorphisms 𝜋_∗ℱ≃𝜋_∗𝜋^∗ℋ≃ℋ⊗𝜋_∗𝒪_𝑌≃⨁_𝑉∈ℐℋ⊗𝑒_𝑉(𝜋_∗𝒪_𝑌); therefore, to understand the 𝐺-structure of 𝜋_∗ℱ it suffices to do it for 𝜋_∗𝒪_𝑌. Thus, we have the next.

Proposition 2.1. Let 𝑋 be a smooth projective variety over an algebraically closed field 𝑘 and 𝜋∶𝑌→𝑋 and étale Galois cover with Galois group 𝐺. Let 𝑉∈ℐ and define ℰ_𝑉 as the 𝒪_𝑋-module 𝜋_∗(𝑉^∗⊗_𝑘𝒪_𝑌)^𝐺=𝑒_𝑈(𝜋_∗(𝑉^∗⊗_𝑘𝒪_𝑌)), where 𝑉^∗ is the dual representation of 𝑉 and 𝑈 is the trivial representation. Then, (1)  𝜋_∗𝒪_𝑌=_𝑉∈ℐ𝑉⊗_𝑘ℰ_𝑉,(2.2) where ℐ is the set of all irreducible representations of 𝐺 over 𝑘 and 𝑟𝑎𝑛𝑘ℰ_𝑉=dim_𝑘𝑉. Also, one has that ℰ_𝑉=𝑒_𝑈(𝑉^∗⊗𝑒_𝑉(𝜋_∗𝒪_𝑌)) and ℰ_𝑉^∗≃ℰ^∗𝑉,(2)  ℰ_𝑉𝑖⊗ℰ_𝑉𝑗≃^𝑟_𝑙=0ℰ^𝑛𝑙_𝑉𝑙,(2.3) where 𝑉_𝑖⊗𝑉_𝑗=⨁^𝑟𝑙=0𝑉^𝑛𝑙_𝑙,(3)each ℰ_𝑉 is simple (i.e., End_𝒪𝑋(ℰ_𝑉)≃𝑘) and ℰ_𝑉𝑖≁ℰ_𝑉𝑗 if 𝑖≠𝑗,(4)for any irreducible representation 𝑉 of 𝐺 one has that ∧^𝑘ℰ_𝑉=ℰ^𝑛𝑖_𝑉𝑖,(2.4) where ∧^𝑘(𝑉)=⨁𝑉^𝑛𝑖_𝑖.

Proof. We recall that 𝑘[𝐺]≃⨁_𝑉∈ℐ𝑉⊗_𝑘𝑉^∗ as 𝑘[𝐺×𝐺]-modules, where actions are given by 𝑔₁×𝑔₂(∑𝜆_𝑔𝑔)=∑𝜆_𝑔𝑔₁⋅𝑔⋅𝑔²⁻¹ and 𝑔₁×𝑔₂(∑𝑣⊗𝛿)=∑𝑔₁(𝑣)⊗𝑔₂(𝛿), respectively. Thus, we have the following 𝐺-invariant isomorphisms: 𝜋^∗_𝑉∈ℐ𝑉⊗_𝑘ℰ_𝑉=_𝑉∈ℐ𝑉⊗_𝑘𝜋^∗ℰ_𝑉≃_𝑉∈ℐ𝑉⊗_𝑘𝑉^∗⊗^𝑘𝒪_𝑌≃𝑘[𝐺]⊗_𝑘𝒪_𝑌≃_𝑔∈𝐺𝑔^∗𝒪_𝑌≃𝜋^∗𝜋_∗𝒪_𝑌.(2.5) Now, Theorem 1 in [6, page 111] asserts that 𝜋^∗ defines an equivalence between the category of locally free 𝒪_𝑋-modules of finite rank and the category of locally free 𝒪_𝑌-modules of finite rank with 𝐺-action. This proves that 𝜋_∗𝒪_𝑌≃⨁_𝑉∈ℐ𝑉⊗_𝑘ℰ_𝑉. In particular, we have that ℰ_𝑉=𝑒_𝑈(𝑉^∗⊗𝑒_𝑉(𝜋_∗𝒪_𝑌)), where 𝑈 is the trivial 𝑘[𝐺]-module.
On the other hand, at the generic point 𝜖 of 𝑋, (𝜋_∗𝒪_𝑌)_𝜖 is the function field 𝐾_𝑌 of 𝑌 which, by the normal basis theorem, is isomorphic to 𝐾_𝑋[𝐺] as 𝐾_𝑋[𝐺]-modules, where 𝐾_𝑋 is the function field of 𝑋. Thus, we have a natural 𝐾_𝑋[𝐺]-isomorphisms: 𝜋_∗𝒪_𝑌_𝜖=_𝑉∈ℐ𝑉⊗_𝑘ℰ_𝑉_𝜖≃𝐾_𝑋[𝐺]=_𝑉∈ℐ𝑉⊗_𝑘𝐾_𝑋^dim𝑉(2.6) and we conclude that ℰ_𝑉 is a locally free sheaf with rank (dim𝑉).
For the last part of (1), we need to see that (ℰ_𝑉𝑖)^∨≃ℰ_𝑉^∨𝑖, but this is a consequence that the trace morphism Tr∶𝜋_∗𝒪_𝑌→(𝜋_∗𝒪_𝑌)^∨ is a 𝐺-invariant isomorphism.
(2) This is immediate from 𝜋_∗𝑉⊗_𝑘𝑊⊗_𝑘𝒪_𝑌^𝐺=𝜋_∗_𝑇∈ℐ𝑇⊗_𝑘𝒪_𝑌^{⟨𝑇,𝑉⊗𝑊⟩}^𝐺=_𝑇∈ℐ𝜋_∗𝑇⊗^𝑘𝒪_𝑌^{⟨𝑇,𝑉⊗𝑊⟩}^𝐺=_𝑉∈ℐℰ^{𝑇⟨𝑇,𝑉⊗𝑊⟩}.(2.7)
(3) We observe that Hom_𝒪𝑋ℰ_𝑉,ℰ_𝑊=Hom_𝒪𝑌𝑉⊗_𝑘𝒪_𝑌,𝑊⊗_𝑘𝒪_𝑌^𝐺=Hom_𝑘[𝐺](𝑉,𝑊),(2.8) and from Schur's Lemma, dim_𝑘Hom_𝑘[𝐺](𝑉,𝑊) is 0 if 𝑉≠𝑊 and 1 if 𝑉≃𝑊 (see [7, page 181]).
(4) We know that tensor functors commutes with the pull back, so we have that 𝜋^∗∧^𝑘ℰ_𝑉≃∧^𝑘𝜋^∗ℰ_𝑉≃∧^𝑘𝑉^∗⊗𝒪_𝑌≃∧^𝑘𝑉^∗⊗𝒪_𝑌≃_𝑇∈ℐ𝑇^∗^{⟨𝑇,∧𝑘𝑉⟩}⊗𝒪_𝑌≃_𝑇∈ℐ𝑇^∗⊗𝒪_𝑌^{⟨𝑇,∧𝑘𝑉⟩}≃_𝑇∈ℐ𝜋^∗ℰ^{⟨𝑇,∧𝑘𝑇𝑉⟩}≃𝜋^∗_𝑇∈ℐℰ^{⟨𝑇,∧𝑘𝑇𝑉⟩},(2.9) and, applying Theorem 1(B) in [6, page 111], we get the desired isomorphism.

Remark 2.2. Note that (4) is valid for each of the Schur functors.

3. Stable Sheaves

Let ℰ be a locally free sheaf over 𝑋 and 𝑚⟼𝜒(ℰ⊗𝒪(𝑚))(3.1) its Hilbert polynomial, where 𝜒(ℰ)=∑(−1)^𝑖ℎ^𝑖(𝑋,ℰ) is the Euler characteristic of ℰ. It is well known that this polynomial can be written as 𝑃(ℰ,𝑚)=^𝑑_𝑖=0𝛼_𝑖(ℰ)𝑚^𝑖𝑖!.(3.2) Define the degree of ℰ as deg(ℰ)=𝛼_𝑑−1(ℰ)−𝑟𝑘(ℰ)𝛼_𝑑−1𝒪_𝑋(3.3) and its slope by 𝜇(ℰ)=deg(ℰ)𝑟𝑘(ℰ),(3.4) where 𝑟𝑘(ℰ) is the rank of ℰ. Recall that a locally free sheaf is 𝜇-𝐻-stable (𝜇-𝐻-semistable) if 𝜇(ℱ)<𝜇(ℰ) for all subsheaf ℱ (𝜇(ℱ)≤𝜇(ℰ), resp.); also a 𝜇-𝐻-semistable locally free sheaf is said 𝜇-𝐻-polystable if it is a direct sum of 𝜇-𝐻-stable sheaves. Any 𝜇-𝐻-stable sheaf is simple, that is, End(ℰ)=𝑘, in particular, a 𝜇-𝐻-polystable sheaf is 𝜇-𝐻-stable if and only if it is a simple sheaf.

The following proposition is proved in [8, pages 62-63] under the assumption of characteristic zero in the base field, but the arguments that surround the proof are valid for any characteristic when we consider only étale covers; however, per clarity we will repeat the proof in Section 5.

Proposition 3.1. Let 𝑋 be a complete variety over 𝑘 and 𝜋∶𝑌→𝑋 be an étale Galois cover, with Galois group 𝐺. Let ℰ be a locally free sheaf on 𝑋. Then, ℰ is 𝜇-𝐻-polystable if and only if 𝜋^∗ℰ is 𝜇-𝜋^∗𝐻-polystable.

Corollary 3.2. Let ℒ be a line bundle on 𝑌; then; for all 𝜇-𝐻-(semi)stable bundle ℱ on 𝑋, ℱ⊗𝜋_∗ℒ is 𝜇-𝐻-(semi)polystable bundle.

Proof. We have that the cover is étale, thus 𝜋^∗(ℱ⊗𝜋_∗ℒ)≃𝜋^∗(𝜋_∗𝜋^∗ℱ⊗ℒ)≃⨁_𝑔∈𝐺𝑔^∗(𝜋^∗ℱ⊗ℒ)≃⨁_𝑔∈𝐺𝜋^∗ℱ⊗𝑔^∗ℒ. Then, by Proposition 3.1, ℱ⊗𝜋_∗ℒ is 𝜇-𝐻-(semi)polystable because 𝜋^∗ℱ⊗𝑔^∗ℒ is 𝜇-𝜋^∗𝐻-(semi)polystable.

Corollary 3.3. Let 𝒢 be a locally free 𝜇-𝜋^∗𝐻-stable 𝒪_𝑌-module. Then, 𝜋_∗𝒢 is 𝜇-𝐻-polystable, and is 𝜇-𝐻-stable if and only if 𝒢≠𝑔^∗𝒢 for all 𝑔≠Id.

Proof. We have that the cover is étale, thus 𝜋^∗𝜋_∗𝒢≃⨁_𝑔∈𝐺𝑔^∗𝒢, so 𝜋_∗𝒢 is 𝜇-𝐻-polystable because 𝑔^∗𝒢≃𝒢 is 𝜇-𝜋^∗𝐻-stable. On the other hand, dimHom𝜋_∗𝒢,𝜋_∗𝒢=ℎ⁰𝑌,_𝑔∈𝐺𝑔^∗𝒢⊗𝒢^∗=_𝑔∈𝐺ℎ⁰𝑌,𝑔^∗𝒢⊗𝒢,(3.5) so 𝜋_∗𝒢 is simple if and only if 𝑔^∗𝒢≠𝒢 for all 𝑔≠1𝑑_𝑌.

Now, we recall the following lemma of the theory of semistable bundles.

Lemma 3.4. Let ℰ be a 𝜇-𝐻-semistable sheaf; if g.c.d(deg(ℰ),𝑟𝑘(ℰ))=1, then ℰ is 𝜇-𝐻-stable.

Proof. If ℰ is not 𝜇-𝐻-stable, then there exists a subsheaf ℱ such that 𝜇(𝐹)=𝜇(ℰ) and 𝑟𝑘(ℱ)<𝑟𝑘(ℰ); thus, we have deg(ℱ)𝑟𝑘(ℰ)=deg(ℰ)𝑟𝑘(ℱ) which contradicts the assumption g.c.d.(deg(ℰ),𝑟𝑘(ℰ))=1.

Corollary 3.5. Let 𝑑,𝑟 be integers such that, 𝑟>0, g.c.d.(𝑑,𝑟)=1=g.c.d.(|𝐺|,𝑟), and suppose that ℒ is a line bundle on 𝑌 such that 𝜋_∗ℒ is 𝜇-𝐻-stable; then, for all 𝜇-𝐻-stable bundle ℱ on 𝑋 of degree 𝑑 and rank𝑟, one has that ℱ⊗𝜋_∗ℒ is 𝜇-𝐻-stable bundle.

Proof. From Proposition 3.1, we have that 𝜋^∗ℱ is 𝜇-𝜋^∗𝐻-polystable with degree 𝑑|𝐺| and rank 𝑟, and from the fact that g.c.d.(𝑑|𝐺|,𝑟)=1 and Lemma 3.4, we have that it is 𝜇-𝜋^∗𝐻-stable. Now, by Corollary 3.3, 𝜋_∗ℒ is 𝜇-𝐻-stable if and only if 𝑔^∗ℒ≠ℒ for all 𝑔∈𝐺; thus, 𝜋^∗ℱ⊗𝑔^∗ℒ≠𝜋^∗ℱ⊗ℒ for all 𝑔∈𝐺, then 𝜋_∗(𝜋^∗ℱ⊗ℒ)≃ℱ⊗𝜋_∗ℒ is 𝜇-𝐻-stable.

Theorem 3.6. Let 𝑋 be a smooth projective variety over an algebraically closed field 𝑘 and let 𝜋∶𝑌→𝑋 be an étale Galois cover with Galois group 𝐺 and 𝜋_∗𝒪_𝑌=_𝑉∈ℐ𝑉⊗_𝑘ℰ_𝑉(3.6) the isotypical decomposition of 𝜋_∗𝒪_𝑌. Then, each ℰ_𝑉 is 𝜇-𝐻-stable with respect to any ample line bundle 𝒪_𝑋(1).

Proof. As 𝜋∶𝑌→𝑋 is étale, we have that 𝜋^∗𝜋_∗𝒪_𝑌≃𝒪^{𝑌⊕|𝐺|}, and from Proposition 3.1, it follows that 𝜋_∗𝒪_𝑌 is 𝜇-𝐻-polystable; now, we just have to see that each ℰ_𝑉 is simple, but this is consequence of Proposition 2.1, Part (3).

Corollary 3.7. Let ℱ be a 𝜇-𝐻-semistable vector bundle on 𝑋, then ℰ_𝑉⊗ℱ is 𝜇-𝐻-semistable for all ℰ_𝑉. Moreover, if ℱ is 𝜇-𝐻-stable, then ℰ_𝑉⊗ℱ is 𝜇-𝐻-polystable.

Proof. We have that 𝜋^∗𝜋_∗𝜋^∗ℱ≃⨁_𝑔∈𝐺𝜋^∗ℱ, so, by Lemma 5.3of Section 5,𝜋_∗𝜋^∗ℱ is 𝜇-𝐻-semistable; now, 𝜋_∗𝜋^∗ℱ≃ℱ⊗𝜋_∗𝒪_𝑌≃⨁_𝑉∈ℐ𝑉⊗ℰ_𝑉⊗ℱ, then ℰ_𝑉⊗ℱ is 𝜇-𝐻-semistable for all 𝑉. Applying Proposition 3.1 we get the second assertion by an analogous argument.

Let ℳ_𝐻(𝑟,𝑑) (ℳ_𝐻(𝑟,𝑑)) be the moduli space parametrizing locally free 𝜇-𝐻-stable sheaves (𝜇-𝐻-polystables, resp.), of rank 𝑘 and degree zero over 𝑘 over 𝑋.

Corollary 3.8. Let 𝑋 be a smooth projective variety over an algebraically closed field 𝑘. Suppose that 𝑋 admits an étale Galois cover with Galois group 𝐺. Then, for all irreducible representation 𝑉 of 𝐺 over 𝑘 and any polarization 𝐻, one has that ℳ_𝐻(dim𝑉,0) is non empty.

Now, on characteristic zero, we have that exterior products of the standard representation of symmetric groups are irreducible (see [9, page 31]); thus, we have the next.

Corollary 3.9. Let 𝑋 be a smooth projective variety over 𝑘, char(𝑘)=0, and let 𝐻 be any polarization. Suppose that the variety admits a Galois cover with Galois group, the symmetric group on d letters 𝔖_𝑑. Then, there exists a nonempty open set 𝑈⊂ℳ_𝐻(𝑑−1,0) such that ⋀^𝑘ℰ is 𝜇-𝐻-stable for all ℰ∈𝑈.

Proof. As the characteristic is zero, the wedge product of 𝜇-𝐻-stables sheaves is 𝜇-𝐻-polystable, in particular 𝜇-𝐻-semistable. So, we have defined a morphism: ℳ_𝐻(𝑑−1,0)⟶ℳ_𝐻⎛⎜⎜⎝⎛⎜⎜⎝𝑑−𝑘1⎞⎟⎟⎠,0⎞⎟⎟⎠(3.7) given by ℰ↦⋀^𝑘ℰ. As the variety ℳ_𝐻(^𝑘𝑑−1,0) is an open set of ℳ_𝐻(^𝑘𝑑−1,0), we only need to see that there is an ℰ∈ℳ_𝐻(𝑑−1,0) such that ⋀^𝑘ℰ is 𝜇-𝐻-stable.
By hypothesis, we assume the existence of an étale Galois cover with Galois group, the symmetric group on 𝑑 letters 𝔖_𝑑; let 𝑉 be the standard representation of 𝔖_𝑑 and ℰ_𝑉 the corresponding sheaf obtained in Theorem 3.6, and recalling that in characteristic zero the exterior algebra of the standard representation ⋀^𝑘𝑉 is irreducible for all 𝑘, we have, from part (4) of Proposition 2.1, that ⋀^𝑘ℰ_𝑉=ℰ_⋀^𝑘_𝑉 is 𝜇-𝐻-stable.

In particular we have the following.

Corollary 3.10. Let 𝑋 be a smooth projective curve of genus 𝑔>1 over a field of characteristic zero. Then, ⋀^𝑘ℰ is 𝜇-𝐻-stable for the generic vector bundle ℰ of degree zero and rank𝑑.

Proof. In this case, the moduli spaces ℳ_𝐻(𝑑,0) and ℳ_𝐻(^𝑑𝑘,0) are irreducible, so we only need to prove the existence of a Galois cover with Galois group 𝔖_𝑑+1 and, by the Lefschetz Principle (see [10]), it suffices to do it for 𝑘=ℂ. Let 𝜋₁(𝑋,𝑥₀)=⟨𝛼₁,𝛽₁,…,𝛼_𝑔,𝛽_𝑔|∏[𝛼_𝑖𝛽_𝑖𝛼^𝑖−1𝛽^𝑖−1]=1⟩ be the fundamental group of 𝑋, with base point 𝑥₀; on the other hand, the symmetric group 𝔖_𝑑+1 is generated by one transposition and one cycle of length 𝑑+1, so we can define a surjective morphism 𝜋₁(𝑋,𝑥₀)→𝔖_𝑑+1, for example, the one given by 𝛼₁↦(1,2), 𝛼₂↦(1,…,𝑑+1), 𝛼_𝑖↦1 if 𝑖>2 and 𝛽₁↦(1,2), 𝛽₂↦(1,…,𝑑+1)⁻¹=(𝑑+1,𝑑,𝑑−1,…,2,1), 𝛽_𝑖↦1 if 𝑖>2.

Proposition 3.11. Let 𝑋 be projective smooth variety over an algebraically closed field 𝑘 and let 𝜋∶𝑌→𝑋 be an étale Galois cover with Galois group 𝐺, and set 𝜋_∗𝒪_𝑌=⨁_𝑉∈𝐼𝑉⊗_𝑘ℰ_𝑉. Let ℱ be a 𝜇-𝐻-stable vector bundle. Then, the following statements are equivalent: (1)𝜋^∗ℱ is 𝜇-𝜋^∗𝐻-stable;(2) Hom(ℱ,ℱ⊗ℰ_𝑉)=0forall𝑉≠𝑉₀;(3)ℱ⊗ℰ_𝑉 is 𝜇-𝐻-stable forall𝑉 and ℱ⊗ℰ_𝑉≠ℱ⊗ℰ_𝑊 for 𝑉≠𝑊.

Proof. (3.9)⇔(2) Since ℱ is 𝜇-𝐻-stable, from Proposition 3.1, we have that 𝜋^∗ℱ is 𝜇-𝜋^∗𝐻-polystable; on the other hand, Hom𝜋^∗ℱ,𝜋^∗ℱ==𝐻𝐻⁰₀𝑌,𝜋^∗ℱ^∨⊗𝜋^∗ℱ=𝐻⁰𝑋,𝜋_∗𝜋^∗ℱ^∨⊗𝜋^∗ℱ=Hom𝑋,ℱℱ,^∨ℱ⊗⊗𝜋_∗𝜋^∗ℱ_𝑉∈𝐼=Homℱ,𝜋_∗𝜋^∗ℱ=Homℱ,ℱ⊗𝜋_∗𝒪_𝑌𝑉⊗_𝑘ℰ_𝑉=_𝑉∈𝐼𝑉⊗_𝑘Homℱ,ℰ_𝑉⊗ℱ.(3.8) Thus, 𝜋^∗ℱ is a simple vector bundle if and only if Hom(ℱ,ℰ_𝑉⊗ℱ)=0 for all 𝑉 different from the trivial representation 𝑉₀.
 (3.9)⇔(3) Notice that 𝜋^∗𝜋_∗𝜋^∗ℱ≃⨁𝑔^∗𝜋^∗ℱ≃𝜋^∗ℱ^⊕|𝐺| is 𝜇-𝜋^∗𝐻-polystable; thus, 𝜋_∗𝜋^∗ℱ≃ℱ⊗𝜋_∗𝒪_𝑌 and ℱ⊗ℰ_𝑉 are 𝜇-𝐻-polystable. Also, we have, from Proposition 3.1, that 𝜋^∗ℱ is 𝜇-𝜋^∗𝐻-polystable; then 𝜋^∗ℱ is simple, and then, 𝜇-𝜋^∗𝐻-stable if and only if dimHom𝜋^∗ℱ,𝜋^∗ℱ^⊕|𝐺|=||𝐺||.(3.9) On the other hand, we have the following relations: Hom𝜋^∗ℱ,𝜋^∗ℱ^⊕|𝐺|==HHoomm𝜋𝜋^∗_∗ℱ,_𝑔∈𝐺𝑔^∗𝜋^∗ℱ=𝐻₀ℱ,𝜋^∗𝜋_∗𝜋^∗ℱ=𝐻₀𝑌,𝜋^∗ℱ^∨⊗𝜋^∗𝜋_∗𝜋^∗ℱ=Hom𝑋𝜋,𝜋_∗∗𝜋^∗ℱ^∨⊗𝜋_∗𝜋^∗ℱ=Homℱ𝜋⊗^∗ℱ𝒪_𝑌,𝜋_∗𝜋^∗ℱ=Homℱ⊗,ℱ_𝑉∈𝐼⊗𝒪_𝑌𝑉⊗_𝑘ℰ_𝑉,ℱ⊗_𝑉∈𝐼𝑉⊗_𝑘ℰ_𝑉=_{𝑉,𝑊∈𝐼}≃_{𝑉,𝑊∈𝐼}Hom𝑉⊗_𝑘ℱ⊗ℰ_𝑉,𝑊⊗_𝑘ℱ⊗ℰ_𝑊Homℱ⊗ℰ_𝑉,ℱ⊗ℰ_𝑊^{(dim𝑉)×(dim𝑊)}≃_𝑉∈𝐼Homℱ⊗ℰ_𝑉,ℱ⊗ℰ_𝑉^(dim𝑉)2_𝑉≠𝑊Homℱ⊗ℰ_𝑉,ℱ⊗ℰ_𝑊^{(dim𝑉)×(dim𝑊)}.(3.10)
Now,|𝐺|=∑_𝑉∈𝐼(dim𝑉)² and, from previous relationships, we have that equality (3.9) is possible if and only if dimHom(ℱ⊗ℰ_𝑉,ℱ⊗ℰ_𝑊) is zero if 𝑉≠𝑊 and ℱ⊗ℰ are simple for all 𝑉. This tested the statement.

Theorem 3.12. Let d,r be integers such that r>0 and g.c.d.(d,r)=1=g.c.d.(|G|,r). Then, for every irreducible representation V of G and ℱ∈ℳ_X(r,d) one has that ℱ⊗ℰ_V is 𝜇-H-stable. In addition, the natural induced morphism ℳ_X(r,d)→ℳ_X(r⋅dimV,d⋅dimV)⊂ℳ_X(r⋅dimV,d⋅dimV) is injective.

Proof. From Proposition 3.1, we have that 𝜋^∗ℱ is 𝜇-𝜋^∗𝐻-polystable with degree 𝑑|𝐺| and rank 𝑟, and from the fact that g.c.d.(𝑑|𝐺|,𝑟)=1 and Lemma 3.4, we have that it is 𝜇-𝜋^∗𝐻-stable. Thus, applying Proposition 3.11, we have that ℱ⊗ℰ_𝑉 is 𝜇-𝐻-stable for any representation 𝑉.
Let us now prove the injectivity of the morphism ℳ_𝑋(𝑟,𝑑)→ℳ_𝑋(𝑟⋅dim𝑉,𝑑⋅dim𝑉).
Let ℱ₁,ℱ₂∈ℳ_𝑋(𝑟,𝑑) and suppose that ℱ₁⊗ℰ_𝑉≃ℱ₂⊗ℰ_𝑉; notice that this isomorphism implies that det(ℱ₁)=det(ℱ₂). From the 𝜇-𝐻-stability of ℱ_𝑖⊗ℰ_𝑉, we have that these vector bundles are simple, so that 1=dimHom(ℱ₁⊗ℰ_𝑉,ℱ₂⊗ℰ_𝑉)=dimHom(ℱ₁,ℱ₂⊗ℰ_𝑉⊗ℰ^∗𝑉). On the other hand, from Proposition 2.1, we have that ℰ^∗𝑉=ℰ_𝑉^∗ and ℰ_𝑉⊗ℰ_𝑉^∗=⨁ℰ^{⟨𝑇,𝑉⊗𝑉∗⟩}_T, so 1=dimHom(ℱ₁⊗ℰ_𝑉,ℱ₂⊗ℰ_𝑉)=⨁_𝑇∈ℐdimHom(ℱ₁,ℱ₂⊗ℰ_𝑇)^{⟨𝑇,𝑉⊗𝑉∗⟩} and we can deduce the existence of a unique representation 𝑇∈ℐ of dimension 1 such that ℱ₁≃ℱ₂⊗ℰ_𝑇. From here we have that det(ℱ₁)≃det(ℱ₂)⊗(ℰ_𝑇)^⊗𝑟, where 𝑟=𝑟𝑘(ℱ₂) and, thus, (ℰ_𝑇)^⊗𝑟≃𝒪_𝑋 and applying part (4) of Proposition 2.1, we obtain that ℰ_𝑇^⊗𝑟=𝒪_𝑋 and so 𝑇^⊗𝑟 is the trivial representation. However, from representation theory, the order of the cyclic group generated by the isomorphism class of 𝑇 should be divided by |𝐺|, but by hypothesis g.c.d.(𝑟,|𝐺|)=1, then 𝑟=1 and 𝑇 is the trivial representation.

4. Examples of Varieties with Group Action: Actions in the Moduli Space of Stable Bundles

Again, let 𝜋∶𝑌→𝑋 be an étale Galois cover with Galois group 𝐺, thus the action of 𝐺 on 𝑌 determines an action on the moduli space ℳ_𝑌(𝑟,𝑑) (ℳ_𝑌(𝑟,𝑑)) of 𝜇-𝐻-(semi)stables sheaves on 𝑌, and from Corollary 3.3, we have a natural morphism ℳ_𝑌(𝑟,𝑑)→ℳ_𝑋(𝑟|𝐺|,𝑑) given by 𝒢↦𝜋_∗𝒢, which factorizes by the quotient ℳ_𝑌(𝑟,𝑑)/𝐺. The aim of this section is the study of such morphism.

Proposition 4.1. Suppose that g.c.d(|𝐺|,𝑑)=1. Then, for all integer 𝑟>0, the natural action of 𝐺 on ℳ_𝑌(𝑟,𝑑) is without fixed points; in particular, for all 𝒢∈ℳ_𝑌(𝑟,𝑑), one has that 𝜋_∗𝒢 is 𝜇-𝐻-stable.

Proof. Let Pic^𝑑𝒪_𝑌(1)(𝑌) be the subvariety of the Picard variety Pic(𝑌) formed by line bundles of degree 𝑑 with respect to 𝒪_𝑌(1)=𝜋^∗𝒪_𝑋(1). Thus, we have defined the determinant morphism det∶ℳ_𝑌(𝑟,𝑑)→Pic^𝑑𝒪_𝑌(1)(𝑌) and this satisfies that det(𝑔^∗(ℰ))=𝑔^∗(det(ℰ)), so it suffices to prove the proposition for Pic^𝑑𝒪_𝑌(1)(Y).
Let ℒ∈Pic^𝑑𝒪_𝑌(1)(𝑌); from Lemma 5.2, we have that deg(𝜋_∗ℒ)=𝑑 and 𝑟𝑘(𝜋_∗ℒ)=|𝐺|, so Lemma 3.4 implies the 𝜇-𝐻-stability of 𝜋_∗ℒ; finally, from Corollary 3.3, 𝑔^∗ℒ≠ℒ for all 𝑔≠Id, then we conclude that there are no fixed points. The last part of the statement is consequence of Corollary 3.3.

Denote by 𝒬^{𝜋𝑟,𝑑} the quotient variety ℳ_𝑌(𝑟,𝑑)/𝐺 and 𝛿∶𝒬^{𝜋𝑟,𝑑}→𝒬^𝜋1,𝑑 the induced morphism by det∶ℳ_𝑌(𝑟,𝑑)→Pic^𝑑(𝑌)=ℳ_𝑌(1,𝑑). The fiber of this map is described in the following.

Corollary 4.2. Let ℒ∈Pic^𝑑(𝑌). Then, if g.c.d(|𝐺|,𝑑)=1, then the restriction of the quotient morphism defines an isomorphism det⁻¹(ℒ)→𝛿⁻¹([ℒ]).

Proof. In order to see this, it suffices to show that 𝑔(det⁻¹(ℒ))∩det⁻¹(ℒ)=∅ which is consequence of the proof of the above proposition.

Corollary 4.3. Let 𝑟 be a positive integer and suppose that g.c.d(|𝐺|,𝑑)=1. Then, one has a natural injective morphism 𝜋_∗∶𝒬^{𝜋𝑟,𝑑}→ℳ_𝑋(𝑟|𝐺|,𝑑)⊂ℳ_𝑋(𝑟|𝐺|,𝑑).

Consider 𝑌 and 𝑋 curves. If g.c.d.(𝑟,𝑑)=1, then we have that ℳ_𝑌(𝑟,𝑑)=ℳ_𝑌(𝑟,𝑑) and that the moduli space is smooth (see [8, Section 4.5]). So, we have the following.

Corollary 4.4. 𝒬^{𝜋𝑟,𝑑} is a smooth subvariety of ℳ_𝑋(𝑟|𝐺|,𝑑).

In general, there is a natural map Pic^𝑑𝒪_𝑌(1)(𝑌)→ℳ_𝑋(|𝐺|,𝑑) whose image is naturally isomorphic to the quotient variety 𝒬^𝜋1,𝑑. A special case is for abelian varieties; we note that, by a theorem of Serre-Lang, every étale cover of an abelian variety 𝑋 has the structure of an abelian variety (see [6, page 167]).

Proposition 4.5. Let 𝑋 be a polarized abelian variety and 𝜋∶𝑌→𝑋 an étale cyclic cover of degree 𝑟. Let 𝑑 be an integer such that g.c.d(𝑟,𝑑)=1. Then, each irreducible component of 𝒬^𝜋1,𝑑 is a smooth irreducible component of ℳ_𝑋(𝑟,𝑑).

Proof. By previous Corollary 4.4, 𝒬^𝜋1,𝑑 is a smooth subvariety of ℳ_𝑋(𝑟|𝐺|,𝑑) and by Proposition 4.1  𝑔^∗ℒ≠ℒ if 𝑔≠Id. Now, let ℒ∈Pic^𝑑(𝑌); thus, 𝑇_𝜋∗ℒℳ_𝑋(𝑟,𝑑) is given by 𝐻⨁¹(𝑋,𝜋_∗ℒ⊗𝜋_∗ℒ⁻¹)≃𝐻¹(𝑋,𝜋_∗(𝜋^∗𝜋_∗𝐿⊗ℒ⁻¹))≃𝐻¹(𝑌,𝜋^∗𝜋_∗ℒ⊗ℒ⁻¹)≃𝐻¹(𝑌,⨁_𝑔∈𝐺𝑔^∗ℒ⊗ℒ⁻¹)≃_𝑔∈𝐺𝐻¹(𝑌,𝑔^∗ℒ⊗ℒ⁻¹); now, by the vanishing theorem for abelian varieties in [6, page 76], we have that ℎ¹(𝐿)=0 for all ℒ∈Pic⁰(𝑌), ℒ≠𝒪_𝑌. Then, 𝑇_𝜋∗ℒℳ_𝑋(𝑟,𝑑)≃𝐻¹(𝑌,𝒪_𝑌)≃𝑇_ℒPic(𝑌)≃𝑇_𝜋∗ℒ𝒬^𝜋1,𝑑.

5. Proof of Proposition 3.1

We will need three lemmas; on them, we will be under the assumptions of Proposition 3.1.

Let us recall an equivalent definition of the degree of an 𝒪_𝑋-module.

Lemma 5.1. For a locally free sheaf ℰ, deg(ℰ)=𝑐₁(ℰ)⋅𝐻^𝑑−1.(5.1) In particular, if ℰ and ℱ are locally free sheaves, then one has deg(ℰ⊗ℱ)=𝑟𝑘(ℰ)deg(ℱ)+𝑟𝑘(ℱ)deg(ℰ).(5.2)

Proof. By Hirzebruch-Riemann-Roch formula, we have 𝑃(ℰ)(𝑚)=𝜒ℰ⊗𝒪_𝑋(𝑚)=degchℰ⊗𝒪_𝑋(𝑚)⋅td(𝒯_𝑋)_𝑛=_𝑘_{𝑖+𝑗=𝑛−𝑘}degch(ℰ)_𝑖⋅td𝒯_𝑋_𝑗⋅𝐻^𝑑−𝑘𝑚^𝑘𝑘!,(5.3) where ch(𝒪_𝑋(𝑚))=∑_𝑘((𝑚𝐻)^𝑘)/𝑘! for some ample divisor 𝐻; thus, 𝛼_𝑑−1(ℰ)==ddeegg𝑐c₁h(ℰ)₁⋅td₀𝒯_𝑋⋅𝐻^𝑑−1+degch₀(ℰ)⋅td₁𝒯_𝑋⋅𝐻^𝑑−1(ℰ)⋅𝐻^𝑑−1+𝑟𝑘(ℰ)2deg𝑐¹𝒯_𝑋⋅𝐻^𝑑−1.(5.4) In particular, for ℰ=𝒪_𝑋, we have 𝛼_𝑑−1𝒪_𝑋=12deg𝑐¹𝒯_𝑋⋅𝐻^𝑑−1(5.5) and the first part of the lemma follows. The second part is consequence of basic properties of Chern classes.

Lemma 5.2. Let ℰ be a locally free 𝒪_𝑋-module. Then, 𝑟𝑘(𝜋^∗ℰ)=𝑟𝑘(ℰ) and deg(𝜋^∗ℰ)=|𝐺|deg(ℰ), in particular 𝜇(𝜋^∗ℰ)=|𝐺|𝜇(ℰ). Let 𝒢 be a locally free 𝒪_𝑌-module. Then, 𝑟𝑘(𝜋_∗𝒢)=|𝐺|𝑟𝑘(𝒢) and deg(𝜋_∗𝒢)=deg(𝒢), in particular |𝐺|𝜇(𝜋_∗𝒢)=𝜇(𝒢).

Proof. Statements about the rank follow from the general theory of finite covers. Now, as an étale cover is affine, we have ℎ^𝑖(𝑌,𝒢)=ℎ^𝑖(𝑋,𝜋_∗𝒢) for any coherent sheaf on 𝑌; thus, 𝑃(𝑌,𝒢)=𝑃(𝑋,𝜋_∗𝒢) and 𝛼_𝑑−1(𝒢)=𝛼_𝑑−1(𝜋_∗𝒢). On the other hand, by Lemma 5.1 we have that deg(𝜋_∗𝒪_𝑌)=𝑐₁(𝜋_∗𝒪_𝑌)⋅𝐻^𝑑−1=𝑐₁(𝜋_∗𝜋^∗𝒪_𝑋)⋅𝐻^𝑑−1=𝜋_∗(𝑐₁(𝜋^∗𝒪_𝑋))⋅𝐻^𝑑−1=|𝐺|𝑐₁(𝒪_𝑋)⋅𝐻^𝑑−1=0, thus 𝛼_𝑑−1(𝜋_∗𝒪_𝑌)=|𝐺|𝛼_𝑑−1(𝒪_𝑋). In general, for a coherent sheaf 𝒢, we have that deg(𝒢)=𝛼_𝑑−1(𝒢)−𝑟𝑘(𝒢)𝛼_𝑑−1(𝒪_𝑌)=𝛼_𝑑−1(𝜋_∗𝒢)−𝑟𝑘(𝒢)|𝐺|𝛼_𝑑−1(𝒪_𝑋)=deg(𝜋_∗𝒢).
Finally, let ℰ be a locally free sheaf on 𝑋, then deg(𝜋^∗ℰ)=deg(𝜋_∗𝜋^∗ℰ)=deg(ℰ⊗𝜋_∗𝒪_𝑌)=𝑟𝑘(𝜋_∗𝒪_𝑌)deg(ℰ)+𝑟𝑘(ℰ)deg(𝜋_∗𝒪_𝑌)=|𝐺|deg(ℰ).

Lemma 5.3. Let ℰ be a locally free 𝒪_𝑋-module. Then, ℰ is 𝜇-𝐻-semistable if and only if 𝜋^∗ℰ is 𝜇-𝐻-semistable.

Proof. If ℰ is not 𝜇-𝐻-semistable, then there is a submodule ℱ with 𝜇(ℱ)>𝜇(ℰ), so, by Lemma 5.2, 𝜇(𝜋^∗ℱ)>𝜇(𝜋^∗ℰ) and 𝜋^∗ℰ is not 𝜇-𝐻-semistable.
For the converse, suppose that 𝜋^∗ℰ is not 𝜇-𝜋^∗𝐻-semistable, then, by the Harder-Narasimhan filtration theorem, there exists a unique submodule 𝒢 such that it is 𝜇-𝜋^∗𝐻-semistable and 𝜇(𝒢)≥𝜇(ℋ) for all submodule of 𝜋^∗ℰ; by the uniqueness, it is invariant under the action of 𝐺, so, by Theorem 1 in [6, page 111], there exists an 𝒪_𝑋-submodule ℱ such that 𝒢≃𝜋^∗ℱ and, by previous lemma, 𝜇(ℱ)>𝐸.

Proof of Proposition 3.1. Suppose that 𝜋^∗ℰ is 𝜇-𝐻-polystable, so by the previous lemma ℰ is 𝜇-𝐻-semistable and by the Jordan-Holder filtration theorem there exists a destabilizing submodule ℱ such that it is 𝜇-𝐻-stable and 𝜇(ℱ)=𝜇(ℰ), so by previous lemmas 𝜋^∗ℱ is 𝜇-𝜋^∗𝐻-semistable and 𝜇(𝜋^∗ℱ)=𝜇(𝜋^∗ℰ) and so it must be a direct summand of 𝜋^∗ℰ. Let 𝜋^∗ℱ→^𝑖𝜋^∗ℰ→^𝑝𝜋^∗ℱ be the inclusion and projection morphisms, so taking the direct image we have 𝐺-invariant morphisms ℱ⊗𝜋_∗𝒪→_𝑌𝑖^∗ℰ⊗𝜋_∗𝒪→_𝑌𝑝^∗ℱ⊗𝜋_∗𝒪_𝑌 with 𝑝_∗∘𝑖_∗=Id_{ℱ⊗𝜋∗𝒪𝑌}, but, now, 𝜋_∗𝒪_𝑌=⨁_𝑉∈ℐ𝑉⊗ℰ_𝑉; hence, we have ℱ⊗𝑉⊗ℰ→_𝑉𝑖^∗ℰ⊗𝑉⊗ℰ→_𝑉𝑝^∗ℱ⊗𝑉⊗ℰ_𝑉 for each irreducible representation 𝑉; in particular for the trivial representation 𝑉₀ we have ℰ_𝑉0=𝒪_𝑋, then ℱ is a direct summand of ℰ.
Now, suppose that ℰ is 𝜇-𝐻-stable; again, by the lemma above, 𝜋^∗ℰ is 𝜇-𝜋^∗𝐻-semistable, so let 𝒢 be a destabilizing submodulo for it. Thus, taking the sum ∑_𝑔∈𝐺𝑔^∗𝒢, we obtain a 𝐺-invariant 𝜇-𝜋^∗𝐻-polystable submodule of 𝜋^∗ℰ which must be the pullback of a 𝜇-𝐻-polystable subsheaf ℱ of ℰ with the same slope, so ℱ=ℰ and then 𝜋^∗ℰ=∑_𝑔∈𝐺𝑔^∗𝒢.